# int 1/(sqrt(x)-root3(x)) dx Use substitution and partial fractions to find the indefinite integral

int1/(sqrt(x)-root(3)(x))dx

Apply integral substitution:u=x^(1/6)

=>du=1/6x^(1/6-1)dx

du=1/6x^(-5/6)dx

du=1/(6x^(5/6))dx

6x^(5/6)du=dx

6(x^(1/6))^5du=dx

6u^5du=dx

int1/(sqrt(x)-root(3)(x))dx=int(6u^5)/(u^3-u^2)du

=int(6u^5)/(u^2(u-1))du

Take the constant out,

=6intu^3/(u-1)du

Integrand is an inproper rational function as degree of numerator is more than the degree of the denominator,

So let's carry out the division,

u^3/(u-1)=u^2+u+1+1/(u-1)

=6int(u^2+u+1+1/(u-1))du

Apply the sum rule,

=6(intu^2du+intudu+int1du+int1/(u-1)du)

Apply the power rule and the common integer:int1/xdx=ln|x|

=6(u^3/3+u^2/2+u+ln|u-1|)

Substitute back u=x^(1/6)

and add a constant C to the solution,

=6(1/3(x^(1/6))^3+1/2(x^(1/6))^2+x^(1/6)+ln|x^(1/6)-1|)+C

=2x^(1/2)+3x^(1/3)+6x^(1/6)+6ln|x^(1/6)-1|+C

=2sqrt(x)+3root(3)(x)+6root(6)(x)+6ln|root(6)(x)-1|+C

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